Chapter 10 - Rotation• Definitions:

– Angular Displacement– Angular Speed and Velocity – Angular Acceleration– Relation to linear quantities

• Rolling Motion• Constant Angular Acceleration• Torque

– Vector directions– Moment Arm

• Newton’s 2nd Law for Rotation• Calculating Rotational Inertia

– Moment of inertia– Using the table– Parallel Axis Theorem– Perpendicular Axis Theorem

• Conservation of Angular Momentum• Rotational Kinetic Energy• Work and Rotational Kinetic Energy

R

Radius vs. position vector

Kinematics Memory Aid

x, x

v, v

a

dx

dt2

2

d xa

dt

vdt

dv

dt adt

Forces cause acceleration!!!

Velocity

• Average velocity

• Instantaneous velocity

2 1

2 1

x xxv

t t t

t 0

x dxv lim

t dt

L

T

Angular Displacement

2 1

Angular Velocity

• Average angular velocity

• Instantaneous angular velocity

2 1

2 1t t t

t 0

dlim

t dt

radians 1

T T

Acceleration

• Average acceleration

• Instantaneous acceleration

2 1

2 1

v v va

t t t

t 0

v dva lim

t dt

2

2

dv d dx d xa

dt dt dt dt

2

L

T

Angular Acceleration

• Average angular acceleration

• Instantaneous angular acceleration

2 1

2 1t t t

t 0

dlim

t dt

2

2

d d d d

dt dt dt dt

2 2

radians 1

T T

Rotational Kinematics Memory Aid

,

,

d

dt

dt2

2

d

dt

d

dt

dt

What causes angular acceleration?

Converting angular to linear quantities

velocity

d dv R R

dt dt

R

tangential acceleration

2 2

2 2

d da R R

dt dt

Radial acceleration

2

R

va

r

2

2R

Ra R

R

tan Ra a a

Frequency vs. angular velocity

• Frequency– Cycles per time interval

– Revolutions per time interval

– Hertz

• Angular velocity– Radians per time interval

– Sometimes called angular frequency

– Radians/sec

2 radf

1 rev

f2

1T

f

Constant Acceleration

0v v at

20 0

1x x v t at

2

2 20 0v v 2a x x

0v vv

2

Constant Angular Acceleration

0 t

20 0

1t t

2

2 20 02

0

2

24.2 rad / s

2rad / s

Problem 1

• A record player is spinning at 33.3 rpm. How far does it turn in 2 seconds.

• The motor is shut off. The record player spins down in 20 seconds (assume constant deceleration).– What is the angular acceleration?– How far does it turn during this coast down?

Vector nature of angular quantities

Rolling without slipping

v R

Problem 2

• A cylinder of radius 12 cm starts from rest and rotates about its axis with a constant angular acceleration of 5.0 rad/s2. At t = 3.0 sec, what is:– Its angular velocity

– The linear speed of the point on the rim

– The radial and tangential components of acceleration of a point on the rim.

Torque causes angular acceleration

• Torque is the moment of the force about an axis

• Product of a force and a lever arm

• Rotational Analog to Newton’s 2nd Law

RF

What if the force is not perpendicular?

RF

R F

RFsin

Vector Multiplication – Cross Product

A B A B sin

ˆ ˆ ˆ ˆ ˆ ˆi i j j k k 0

ˆ ˆ ˆi j k

x y zˆ ˆ ˆA A i A j A k

x y zˆ ˆ ˆB B i B j B k

x y z

x y z

ˆ ˆ ˆi j k

A B A A A

B B B

ˆ ˆ ˆj k i ˆ ˆ ˆk i j

Right Hand Rule II

x y z

x y z

ˆ ˆ ˆi j k

C A B A A A

B B B

Vector Multiplication – Scalar Product

A B A B cos

ˆ ˆ ˆ ˆ ˆ ˆi i j j k k 1

ˆ ˆ ˆ ˆ ˆ ˆi j i k j k 0

x y zˆ ˆ ˆA A i A j A k

x y zˆ ˆ ˆB B i B j B k

x x y y z zA B A B A B A B

The Torque Vector

R F

R F sin

R

R

Problem 3

• Find the net torque on the wheel about the center axle

Rotational Inertia

R F sin R ma

2R mR mR

I

2I mRMoment of inertia for a single particle

General Moment of Inertia

n2

i ii 1

I m R

2I R dm

Problem 4

• Three equal point masses are rotating about the origin at 2 rad/sec.

• The masses are located at (4m, 0) (0, 4m) and (4m, 4m).

• Each mass is 2 kg

• Find the moment of inertia.

Moment of inertia of a uniform cylinder

2I R dm dm dV RdRd dz

0R 2 z 42 20

0

0 0 0

R 1I R RdRd dz 2 z MR

4 2

See Figure in book

Moments of Inertia of various objects

If particular axis is not in the table,use the parallel axis theorem:

2PI I MR

Problem 5

• A disk with radius, R, and mass, M, is free to rotate about its axis. A string is wrapped around its circumference with a block of mass, m, attached. This block is released from rest and falls.

• Find the tension in the string

• Find the acceleration

• Find the velocity after the mass has fallen a distance, h.

m

R

M

Angular Momentum

p mv

L I

dpF ma

dt

dLI

dt

If there are no torques:

0 0L I I constant

Two conservation of angular momentum demonstrations

Precession

Kepler’s 2nd Law

• The Law of Areas– A line that connects a

planet to the sun sweeps out equal areas in equal times.

1dA rvdt

2

2 vL I mr mrv Constant

r

1 LdA dt

2 m

Rotational Kinetic Energy

21K mv

2

W F d

dW dP

dt dt

dW F dP F v

dt dt

21K I

2

W d

2 2f i

1 1W I I

2 2

Problem 6 - Energy

• A disk with radius, R, and mass, M, is free to rotate about its axis. A string is wrapped around its circumference with a block of mass, m, attached. This block is released from rest and falls.

• Find the tension in the string

• Find the acceleration

• Find the velocity after the mass has fallen a distance, h.

m

R

M

Sphere rolling down a hill

Find the velocity at the bottom of the hill?

Mass = M, initially at rest

Which is fastest?